One bottleneck in large-scale genome sequencing projects is reconstructing the full genome sequence from the short subsequences produced by current technologies. The final stages of the genome assembly process inevitably require manual inspection of data inconsistencies and could be greatly aided by visualization. This paper presents our design decisions in translating key data features identified through discussions with analysts into a concise visual encoding. Current visualization tools in this domain focus on local sequence errors making high-level inspection of the assembly difficult if not impossible. We present a novel interactive graph display, ABySS-Explorer, that emphasizes the global assembly structure while also integrating salient data features such as sequence length. Our tool replaces manual and in some cases pen-and-paper based analysis tasks, and we discuss how user feedback was incorporated into iterative design refinements. Finally, we touch on applications of this representation not initially considered in our design phase, suggesting the generality of this encoding for DNA sequence data.